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**Lectures on symplectic geometry.**
*(English)*
Zbl 1016.53001

Lecture Notes in Mathematics. 1764. Berlin: Springer. xii, 217 p. EUR 32.95/net; sFr. 55.00; £23.00; $ 45.80 (2001).

More or less the symplectic geometry grew out of developments in classical mechanics and this can be seen in these lectures as well. Indeed, the chapter on Hamiltonian mechanics and the subsequent ones: Moment maps and symplectic relations occupy more than one third of the volume of the book. Before that the author presents the classical material about symplectic vector spaces, cotangent bundles and the fundamental notion and object in this field – the so called Lagrangian submanifold, along with the proof of the local structure theorem. This theorem says roughly that general symplectic manifold can be considered as a cotangent bundle of another manifold of half dimension.

Being even-dimensional, by their very definition the symplectic manifolds possess odd-dimensional counterparts known as contact manifolds. Besides, there are well defined procedures how to pass from one category to the other one known as contactization, respectively symplectization. All this together occupies another third of the book. The rest is devoted to the compatibility of symplectic and complex structures (when the latter one exists) and to the study of a special class of manifolds known as Kählerian which lies in the intersection of complex, Riemannian and symplectic geometries.

As a whole the book is well designed and can serve for both purposes – as an introduction and a good source of references on the subject. The only drawback is the discrepancy in the numbering of references. In most cases a shift by two works but not in general.

Being even-dimensional, by their very definition the symplectic manifolds possess odd-dimensional counterparts known as contact manifolds. Besides, there are well defined procedures how to pass from one category to the other one known as contactization, respectively symplectization. All this together occupies another third of the book. The rest is devoted to the compatibility of symplectic and complex structures (when the latter one exists) and to the study of a special class of manifolds known as Kählerian which lies in the intersection of complex, Riemannian and symplectic geometries.

As a whole the book is well designed and can serve for both purposes – as an introduction and a good source of references on the subject. The only drawback is the discrepancy in the numbering of references. In most cases a shift by two works but not in general.

Reviewer: Ivailo Mladenov (Sofia)

### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

53Dxx | Symplectic geometry, contact geometry |

53D20 | Momentum maps; symplectic reduction |

53D05 | Symplectic manifolds (general theory) |

53D12 | Lagrangian submanifolds; Maslov index |

53D10 | Contact manifolds (general theory) |

53D15 | Almost contact and almost symplectic manifolds |